Completeness in supergravity constructions

TL;DR

This paper proves that certain geometric maps in supergravity preserve completeness, leading to a classification of complete quaternionic Kahler manifolds in low dimensions and providing new examples.

Contribution

It establishes the preservation of completeness under supergravity r- and c-maps and classifies low-dimensional complete quaternionic Kahler manifolds derived from hypersurfaces.

Findings

Supergravity r- and c-maps preserve completeness.

Classification of all complete quaternionic Kahler manifolds up to dimension 12.

Construction of new complete examples in 16 dimensions.

Abstract

We prove that the supergravity r- and c-maps preserve completeness. As a consequence, any component H of a hypersurface {h=1} defined by a homogeneous cubic polynomial such that -d^2 h is a complete Riemannian metric on H defines a complete projective special Kahler manifold and any complete projective special Kahler manifold defines a complete quaternionic Kahler manifold of negative scalar curvature. We classify all complete quaternionic Kahler manifolds of dimension less or equal to 12 which are obtained in this way and describe some complete examples in 16 dimensions.